Questions tagged [cauchy-product]

For questions about the Cauchy product, the discrete convolution of two sequences.

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41 views

What is the Laurent series of $e^{z+z^{-1}}$? (Detailed derivation using Cauchy product of series)

We know that: $$e^{z+z^{-1}}=\underbrace{\left(\sum_{n=0}^{\infty}\frac{z^n}{n!}\right)\cdot\left(\sum_{m=0}^{\infty}\frac{z^{-m}}{m!}\right)}_{=:\alpha}.$$ Now I am having trouble, applying the ...
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58 views

What is the difference between a double infinite series and the Cauchy Product?

Let's say we have the double infinite series $\sum\limits_{n=1}^\infty \sum\limits_{m=1}^\infty a_n b_m$ which is absolutely convergent. Furthermore, the two series $\sum\limits_{n=1}^\infty a_n$ and $...
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1answer
101 views

Structure of product of two polynominals $\left( \sum_{k=0}^{n} a_k x^k \right) \cdot \left( \sum_{l=0}^{m} b_l y^l \right)$

I am interested in the product $$ z = f(x) \cdot g(y) = \left( a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \right) \cdot \left( b_0 + b_1 y + b_2 y^2 + \cdots + b_m y^m \right) $$ With $n = 1$, $m = 2$ ...
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1answer
50 views

What are other elegant counterexamples to the convergence of a Cauchy product?

What is an elegant case of two convergent series $\sum a_n$ and $\sum b_n$ which Cauchy product $$\sum_{n=0}^{\infty} c_n = \sum_{n=0}^\infty \sum_{k=0}^n a_kb_{n-k}$$ diverges? The most common, I ...
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2answers
84 views

How do you prove the Cauchy product (multiplication) of two infinite power series (generating functions) which have different exponents/indices?

I'm trying to multiply 2 generating functions ( (1/(1-x) ) and ( 1/(1-x^5) ) which have different denominations so that I can find the coefficient. When browsing math.stackexchange I found the ...
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1answer
70 views

An expansion for $\ln^2\Gamma(x+1)$

I was reading Irresistible Integrals by Victor H. Moll, where I encountered the following Taylor series expansion of $\ln\Gamma(1+x)$ $$ \ln\Gamma(1+x)=-\gamma x + \sum_{k=2}^{\infty}\dfrac{(-1)^k \...
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38 views

Applying Cauchy product to $e^x$ and $\cos(x)$

Let's consider the improper integral $\int_0^{\infty}e^{-ax}\cos(bx)dx$. I know that we can evaluate it by using integration by parts. The result is: $$ \int_0^{\infty}e^{-ax}\cos(bx)dx=\frac{a}{a^2+b^...
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1answer
61 views

Why is the Cauchy product of the series $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$ diverging?

I'm new to series and I'm reading old questions here to see some examples of tests for divergence and convergence. In the following, I have a question about the conclusion: https://math.stackexchange....
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72 views

Double series and Cauchy product in Hilbert space

Let $(\delta_{i})_{i \geqslant 0}$ be a canonical orthonormal basis in $\ell^{2}\left(\mathbb{N}_{0}\right)$. Does double series $$\sum\limits_{i \geq 0}\sum\limits_{k \geq 0} c_{i}\tau_{k}\delta_{i + ...
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66 views

Proving a summation formula using the general Leibniz rule

I am trying to prove the following relations: $$ \partial^{N}(f^{N+1}g) =\sum_{n+m=N}\frac{N!}{n!\,(m+1)!}\,\big[\partial^{n}(f^{n}g)\big]\,(\partial^{m}f^{m+1}), \qquad N\geq0, $$ and $$ \frac{1}{N+2}...
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1answer
47 views

Can we write $\sum_{k=n}^\infty\sum_{\ell=0}^k a_\ell b_{k-\ell}$ as a product of series?

We have the Cauchy product $$ \sum_{k=0}^\infty a_k\sum_{\ell=0}^\infty b_\ell=\sum_{k=0}^\infty\sum_{\ell=0}^k a_\ell b_{k-\ell}. $$ Do we have a way of writing $$ \tag{1} \sum_{k=n}^\infty\sum_{\ell=...
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1answer
57 views

Cauchy series "Product Like"

If I have: $$\sum_{q=0}^{+\infty}\alpha_q \sum_{r=0}^q \beta_r\gamma_{q-r}$$ Is there a way to manipulate this expression in order to simplify it or at least rewrite into another form? I've just ...
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1answer
29 views

Cauchy product of alternating geometric series

I want to prove that for $ -1\lt q_1 \le q_2 \lt 1 $ then the following holds: $$\sum_{n=0}^{\infty}(-1)^n \sum_{k=0}^n q_1^k \cdot q_2^{n-k} = \frac1{(1+q_1)(1+q_2)}$$ By the first inequality, $ -1\...
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1answer
81 views

Using Cauchy product for an integral

To evaluate $$ \int_0^1 e^x \ln(x+1)dx $$ I was thinking about using the cauchy product of the taylor series of $e^x$ and $\ln(x+1)$. We know that $$ e^x = \sum_{n=0}^\infty \frac{x^n}{n!} $$ and $$ \...
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23 views

Conditions for the Following Series to Converge

The problem is to show on which conditions the following series converge $$ \sum_{n=0}^\infty\left({\sum_{k=1}^{n}a^k}{(-1)^{(n-k)+1}\frac{(n-k)^2}{e^{n-k}}} \right) $$ I tried to simplify the series ...
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1answer
82 views

Showing that $2C(x)^2 = C(2x)+1$ using Cauchy product Formula

I'm stuck trying to calculate $2C(x)^2 = C(2x)+1$ with $C(x) =\sum_{n=0}^{\infty} (-1)^n \frac {x^{2n}} {(2n)!}$ (Cosine Power series) So I've seen proofs for this identity before, but never using the ...
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38 views

Commutative monoid structures on $\Bbb{N}$

Suppose $m \oplus n$ is a commutative and associative binary relation $\oplus: \Bbb{N} \times \Bbb{N} \to \Bbb{N}$, and that $1$ is an identity element for this operation. In other words, $(\oplus, \...
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1answer
50 views

Showing $\left(\sum\limits_{j=0}^n a_jx^j\right)\left(\sum\limits_{k=0}^n b_kx^k\right)=...$ for certain properties

Show that $\left(\sum\limits_{j=0}^n a_jx^j\right)\left(\sum\limits_{k=0}^n b_kx^k\right)=\sum\limits_{m=0}^{2n}\left(\sum\limits_{j+k=m} a_jb_k\right)x^m=\sum\limits_{m=0}^{2n}(a_0b_m+a_1b_{m-1}+...+...
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1answer
44 views

How to prove this series Cauchy product?

Let the first series be $\sum_{k=0}^{\infty} u_{k}(x-a)^{k}$. Let the second series be $\sum_{k=0}^{\infty} v_{k}(x-a)^{k}$ Their convergence areas are $R_{1}$ and $R_{2}$. I don't know how to ...
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2answers
177 views

Cauchy product summation converges

I had a previous question here, which I'm quoting: How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1) (n-k+1)}$$ I tried to prove ...
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1answer
56 views

Cauchy Product of two geometric series.

So I wanted to calculate/prove that $$\frac{1}{(1-q)^2} = \displaystyle\sum_{n=0}^{\infty}(n+1)q^n,$$ $q\in \mathbb{C},$ $\mid q \mid < 1$ using the Cauchy-Product. So this is my solution and I'm ...
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1answer
36 views

show the power series converges for |x|<1, and find the closed forms

\begin{matrix} 1) S(x) = \sum_{n=0}^\infty (1+ \frac{1}{1!} + \frac{1}{2!} + ...+ \frac{1}{n!}) x^n \\ 2) S(x) =\sum_{n=1}^\infty a_n x^n, a_n= \Big\{ \begin{alignedat}{3} -2/n ,n=3k \\ 1/n,n \neq 3k ...
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1answer
114 views

Implementation of Cauchy product on $\cos x\cdot \sin x$

I have a mistake that I can't find somewhere along the way. Please help me find the place things go wrong. Find the product of $\cos x$ and $\sin x$ as defined: $$\cos(x) = \sum_{k=0}^{\infty} \frac{\...
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36 views

Cauchy product of 2 series

Let $z \in \mathbb{C}$. I want to prove that $$\sum_{n\in\mathbb{Z}} \frac{(-1)^n}{z-n} \sum_{n\in\mathbb{Z}} \frac{(-1)^n}{z-n} = \sum_{n\in\mathbb{Z}} \frac{1}{(z-n)^2}.$$ Using the Cauchy Product ...
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1answer
44 views

How to show that these sums are equal?

Assuming that the product is associative, I would like to show that $$ \sum_{s=0}^{k} (\sum_{i=0}^{s} (\alpha_i \cdot \beta_{s-i}) \cdot \gamma_{k-s}) = \sum_{s=0}^{k} ( \sum_{i=0}^{k-s} \alpha_s \...
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3answers
129 views

Product of two generating functions in terms of the ordinary generating function? (recurrence relation solving)

I have the following recurrence relation that I am trying to solve: $a_0 = 1$ $$ a_n = \sum_{i = 0}^{n-1} (i+1)a_i$$ for $ n \geq 1$. Let's define $A(x) = \sum_{n= 0}^\infty a_nx^n$, so multiplying ...
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1answer
47 views

Cauchy product for power series of different powers

How would you find the Cauchy product of two power series of different powers? For example, I want to find the Cauchy product of the two series $ \exp(x) = \sum_{k=0}^{\infty}\frac{x^k}{k!}$ and $ \...
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2answers
86 views

An example: a convergence series, a divergent series, whose Cauchy product is convergent.

How to find an example: a convergence series $\sum a_n$, a divergent series $\sum b_n$, whose Cauchy product $\sum c_n$ with $c_n=\sum_{i+j=n}a_ib_j$ is convergent? Is there a simple example?
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76 views

The Cauchy Product : Calculate the coefficients, $c_n$, in the product.

I found this definition to be different from other ones and the answer makes me feel weird. If $$A_{N}(x)=\sum_{n=0}^{n=N}a_{n}x^n$$ and $$B_{N}(x)=\sum_{n=0}^{n=N}b_{n}x^n,$$ calculate the ...
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1answer
172 views

Integrating the square of an infinite series

Just out of plain curiosity, I want to know how to evaluate the integral of the square of an infinite series. For example, if $$f\left(x\right)=\sum_{n=0}^{\infty}c_n\left(x-a\right)^{n},$$ where $c_n$...
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238 views

Are the unconditionally convergent series, with terms in a Banach algebra, closed under the Cauchy product?

We have a Banach algebra $\mathbb L$, and two sequences $(A_0,A_1,A_2,\cdots),\;(B_0,B_1,B_2,\cdots)\in\mathbb L^{\mathbb N}$, for which the sums $\sum_{n\in\mathbb N}A_n,\;\sum_{n\in\mathbb N}B_n$ ...
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303 views

The necessity of absolute convergence in the convergence of the Cauchy product of series?

The Mertens' theorem claims that Suppose $\sum_{n=0}^\infty a_n,\sum_{n=0}^\infty b_n$ are two convergent series of complex numbers, convergent to $A,\beta$ respectively. If $\sum_na_n$ converges ...
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1answer
178 views

Is the sequence space $\ell^p$ closed under the Cauchy product?

Given that two formal power series have coefficients in $\ell^p$, $$a(x)=\sum_{n=0}^\infty a_nx^n,\quad\sum_{n=0}^\infty|a_n|^p<\infty$$ $$b(x)=\sum_{n=0}^\infty b_nx^n,\quad\sum_{n=0}^\infty|b_n|...
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80 views

Is there a way to re-write $\sum_{n=1}^{\infty}{\left(\sum_{k=1}^{n}{\frac{1}{k}}\right)}z^n$

Is there a way to re-write $\sum_{n=1}^{\infty}{\left(\sum_{k=1}^{n}{\frac{1}{k}}\right)}z^n$ I was thinking that I could use the cauchy product, but therefore I have to achieve this:$$\left(\sum_{...
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2answers
44 views

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$ I believe that I have to use the cauchy product? But how do transform the expression to be a product ...
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1answer
210 views

Product of Two Summations with Different Upper Limits

I'm trying to multiply two different finite summations with different upper limits. I've tried Cauchy Product but i think it's valid for same upper limits. I also tried to split the summation. Any ...
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3answers
58 views

Find the power series of $f(x)=\frac{x}{x+1}$ in $x_0=0$

In 1st Semester Calculus book I found an exercise that asks me to find the above power series of the function at the point $x_0 = 0$ using the geometric series formula and the Cauchy-Product. So far I'...
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1answer
212 views

Cauchy product and geometric series [duplicate]

I was given this series: Let $q \in \mathbb{C}, \mid q\mid <1. $ $$\frac{1}{2}\sum_{n=0}^{\infty} (n^2 +3n +2)q^n $$ Now I have to show that $$\frac{1}{(1-q)^3}=\frac{1}{2}\sum_{n=0}^{\infty} (...
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1answer
794 views

Multiplying two polynomials: explanation of the general formula for the coefficients

If $$f(x)=a_nx^n+...+a_0$$ and $$g(x)=b_mx^m+…+b_0$$ then $$f(x)\cdot g(x)=c_{m+n}x^{m+n}+...+c_0$$ where $c_k=\sum_{r+s=k}a_rb_s,\quad k=0,....,m+n$ I know that the degree $0$ and $(m+n)$ exists, ...
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2answers
187 views

Generating Function For Catalan Numbers Type Sequence

I've been working my way through an old post, but I don't think the solution offered can be correct. The question is; Find the generating function (within a choice of sign) for: $$c_{n+1} = 2\sum_{k=...
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69 views

Cauchy product of power series with $x^{2n}$

I am trying to rewrite a function $y(x)=\frac{1}{1+x+x^2+x^3}$ as a series. I used geometric series and got to two power series that I need to Cauchy product, however, one of them has $x^{2n}$ and I ...
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68 views

Cauchy product for the reciprocal of the polynomial $x - 2x^2 + 3x^3 - 4x^4$

I have come across some Laurent series in which the denominator of a fraction contains a power series. Looking around I came across this Calculate Laurent series for $1/ \sin(z)$ which suggests that ...
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1answer
35 views

How was Merten's theorem proven on the step comparing the convergence of one sequence and the Cauchy product partial sum?

I'm trying to understand step 3 of the proof of Merten's theorem. How is it known that $N*_{sup}|B_i-B|$ term converges slow enough that there is an N where equation 3 is true: $|a_n|\le\frac{\...
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2answers
524 views

Proving $\sin(2x) = 2\sin(x)\cos(x)$ using Cauchy product

I've stumbled upon this question. I can't understand why all even terms of the Cauchy product are $0$, since we add only positive numbers: \begin{equation*} c_n = \begin{cases} \sum\limits_{k=0}^{m} \...
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3answers
237 views

On the series expansion of $\frac{\operatorname{Li}_3(-x)}{1+x}$ and its usage

Recently I have asked about the evaluation of an integral involving a Trilogarithm $($which can be found here$)$. Pisco provided a quite an elegant approach starting with a functional equation of the ...
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2answers
103 views

Simplifying $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{l=0}^{\min(n,m)}a_{l}b_{m-l}c_{n-l}$

I have come across a sum of the following form; $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{l=0}^{\min(n,m)}a_{l}b_{m-l}c_{n-l}$$ and want to simplify it (in particular to remove the $min(n,m)$). ...
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0answers
462 views

Series whose Cauchy product is absolutely convergent - A general example

Is there series that is divergent or conditionally convergent with absolutely convergent Cauchy product? Seems like there is a group of these examples! Perhaps finding divergent series with ...
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1answer
556 views

Conditionally converges $\sum_{k=0}^\infty a_n$, $\sum_{k=0}^\infty b_n$ and also their Cauchy product

Edited: I post a new post here that is somewhat related to this question. It proved that: If $\sum_{k=0}^\infty a_n$ and $\sum_{k=0}^\infty b_n$ converges conditionally (not absolutely), then their ...
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1answer
68 views

Calculating the limit based on the cauchy product

I'm attempting to find the limit of $$ \sum^\infty_{k=0} k^2 q^k$$ using the result of the cauchy product $$ \sum^\infty_{k=0} k q^k * \sum^\infty_{k=0} q^k$$ and I have calculated the cauchy ...
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3answers
157 views

Summation Recurrence Relation

How to solve this Summation Recurrence Relation: $$x_n=\sum_{i=1}^n a_ix_{n-i}\,,\,\,\,n\ge1$$where, $x_0=1$ and $a_n$ is some arbitrary sequence. The right hand side of the recurrence looks ...